A global solution curve for a class of semilinear equations

We use bifurcation theory to give a simple proof of existence and uniqueness of a positive solution for the problem $$ Delta u - lambda u+u^p = 0 quad mbox{for } |x| < 1, quad u = 0 quad mbox{on } |x| = 1, $$ where $x in {mathbb R}^n$, for any integer $n geq 1$, and real 1 less than $p (n+2)/(n-2)$, $lambda geq 0$. Moreover, we show that all solutions lie on a unique smooth curve of solutions, and all solutions are non-singular. In the process we prove the following assertion, which appears to be of independent interest: the Morse index of the positive solution of $$ Delta u +u^p = 0 quad mbox{for } |x| < 1, quad u = 0 quad mbox{on } |x| = 1 $$ is one, for any 1 less than p less than $(n+2)/(n-2)$.